Advancement is not much of the game in Fate.
Many games are intended for campaign play, where you start with a fraction of the power you might eventually aspire to.
In Fate Core and Fate Accelerated, you charge out of the gate as competent, powerful characters capable of making dramatic change in the world. A one-shot is enough to experience very significant parts of the game.
But to address your specific concerns:
The aspects of refresh that might come up in Fate Accelerated campaign play are:
Getting more Fate Points than your refresh value, and restricting yourself from spending them to start the next session with more.
Getting more Refresh at major milestones.
Spending Refresh to get additional stunts beyond your free initial three.
That first one is more of a theoretical concern. I have never seen a game where people had anything near their starting number of Fate Points at the end of it.
The second and third are, yes, basically how character development happens in Fate, but you're going to be feeling your character out in your first session, and especially if you use the quick start rules, you're going to have some character-development-like moments along the way. (I'll summarize the quick start rules at the end.)
The aspects of consequences that might come up in Fate Accelerated campaign play are:
- Taking a lesser consequence than you might be inclined to because consequences stick around for much longer the more Stress they absorb.
...that's it, pretty much.
But the thing is, that's not really a big advantage, because of one thing: consequences are real. When you take a consequence, you have to write down what it means, and the more severe a consequence is, the more it impacts you. It's basically a negative Aspect, and enemies get a free hit on it.
So, like, the only scenario where it really matters is if you're about ready to wrap for the session anyway and you take a bigger consequence that nobody's going to even take advantage of.
The quick start rules are linked at the Fate SRD here. For Fate Accelerated you can use them mostly as written, but a couple of things will need to change.
- instead of picking your best skill, pick your high (+3) and low (+0) approaches. Fill in the middles as you like during play. Ideally everyone should wind up with different strengths and weaknesses from one another.
- you start with three Fate Points and can get up to three Stunts for free, without spending anything. Probably cap there for the first session.
In that way you can spend as much of the single session as possible playing the game, essentially doing character creation under yourself as you go.